The curse of dimensionality looms over many studies in science and engineering. Low-order systems provide conceptual clarity but often fail to reveal the extent of possible complexity, whereas high-order systems present a host of daunting challenges to the analyst, not least the classification and visualization of typical behavior. In this paper, we detail the behavior of systems that fall somewhere between a classification of low- and high-order.[Formula: see text][Formula: see text]We present both theoretical and experimental investigations into the nonlinear behavior of a couple of mechanical systems with three mechanical/structural degrees-of-freedom (DOF), with a special focus on bifurcation and multiple equilibria. Useful insight is provided by observation of transient trajectories as they meander about and between equilibria, especially revealing the influence of unstable equilibria, not normally accessible in an experimental context. For instance, the influences of index-1 saddles are mainly detected in three aspects: determining the systems capability to snap-through by generating accessible snap-though tubes, attracting nearby trajectories temporarily oscillating around it, and separating adjacent trajectories. Iso-potentials are 3D-printed to present the energy landscape. For these systems, the 3D configuration space allows considerable complexity, but is also somewhat amenable to geometric interpretation. By varying a mass/stiffness ratio as a control parameter, bifurcation structures and morphing potential energy landscapes exhibiting up to 11 equilibria are obtained. Finally, analytical and experimental studies reveal that parametric excitations can stabilize some unstable equilibria under the right amplitudes and frequencies.

中文翻译：

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离散机械系统的稳定性和分岔：一项实验和分析研究

维度的诅咒笼罩在科学和工程领域的许多研究中。低阶系统提供了清晰的概念，但通常无法揭示可能的复杂程度，而高阶系统则给分析师带来了许多艰巨的挑战，尤其是典型行为的分类和可视化。在本文中，我们详细介绍了介于低阶和高阶分类之间的系统行为。[公式：见正文][公式：见正文]我们对一对非线性行为进行了理论和实验研究具有三个机械/结构自由度 (DOF) 的机械系统，特别关注分叉和多重平衡。观察瞬态轨迹可以提供有用的洞察力，因为它们在平衡点周围和平衡点之间蜿蜒曲折，特别是揭示了不稳定平衡的影响，在实验环境中通常无法获得。例如，index-1 鞍座的影响主要在三个方面进行检测：通过生成可访问的snap-through 管来确定系统的snap-through 能力，吸引附近临时在其周围振荡的轨迹，以及分离相邻的轨迹。等电位是 3D 打印的，以呈现能源景观。对于这些系统，3D 配置空间允许相当大的复杂性，但也有点服从几何解释。通过改变质量/刚度比作为控制参数，获得了表现出多达 11 个平衡的分叉结构和变形势能景观。最后，